- FindStat

Identifier

Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001617: Lattices ⟶ ℤ

Values

([],1) => ([],1) => 0
([],2) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => 1
([],3) => ([],1) => 0
([(1,2)],3) => ([(0,1)],2) => 1
([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],4) => ([],1) => 0
([(2,3)],4) => ([(0,1)],2) => 1
([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],5) => ([],1) => 0
([(3,4)],5) => ([(0,1)],2) => 1
([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([],6) => ([],1) => 0
([(4,5)],6) => ([(0,1)],2) => 1
([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([(0,5),(1,4),(2,3)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([],7) => ([],1) => 0
([(5,6)],7) => ([(0,1)],2) => 1
([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(3,6),(4,5)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,6),(4,5),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(2,3),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
([(1,6),(2,5),(3,4)],7) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

Description

The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice

$L$ is a function

$v:L\mapsto\mathbb R$ satisfying

$v(a\vee b) + v(a\wedge b) = v(a) + v(b).$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient Mp00196The modular quotient of a lattice..
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Map

connected vertex partitions

Description

Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph

$G = (V,E)$ is a set partition of

$V$ such that each part induced a connected subgraph of

$G$ . The connected vertex partitions of

$G$ form a lattice under refinement. If

$G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on

$n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.